| L(s) = 1 | − 2-s − 3-s + 2·4-s + 5·5-s + 6-s − 5·10-s − 6·11-s − 2·12-s − 2·13-s − 5·15-s − 8·17-s − 4·19-s + 10·20-s + 6·22-s + 10·23-s + 10·25-s + 2·26-s − 8·29-s + 5·30-s + 11·32-s + 6·33-s + 8·34-s − 15·37-s + 4·38-s + 2·39-s + 4·43-s − 12·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 4-s + 2.23·5-s + 0.408·6-s − 1.58·10-s − 1.80·11-s − 0.577·12-s − 0.554·13-s − 1.29·15-s − 1.94·17-s − 0.917·19-s + 2.23·20-s + 1.27·22-s + 2.08·23-s + 2·25-s + 0.392·26-s − 1.48·29-s + 0.912·30-s + 1.94·32-s + 1.04·33-s + 1.37·34-s − 2.46·37-s + 0.648·38-s + 0.320·39-s + 0.609·43-s − 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6483017256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6483017256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + T - T^{2} - 3 T^{3} - T^{4} - 3 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 6 T + 5 T^{2} - 6 T^{3} + 49 T^{4} - 6 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 49 T^{4} + 16 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 8 T + p T^{2} + 60 T^{3} + 461 T^{4} + 60 p T^{5} + p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 4 T - 3 T^{2} + 62 T^{3} + 605 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - 10 T + 37 T^{2} - 200 T^{3} + 1389 T^{4} - 200 p T^{5} + 37 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 8 T + 5 T^{2} - 8 p T^{3} - 59 p T^{4} - 8 p^{2} T^{5} + 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 9 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 15 T + 63 T^{2} + 65 T^{3} + 144 T^{4} + 65 p T^{5} + 63 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 31 T^{2} + 180 T^{3} + 1501 T^{4} + 180 p T^{5} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 2 T - 23 T^{2} + 250 T^{3} + 2601 T^{4} + 250 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 5 T - 43 T^{2} - 5 p T^{3} + 1244 T^{4} - 5 p^{2} T^{5} - 43 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 408 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 2 T - 57 T^{2} - 64 T^{3} + 3905 T^{4} - 64 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 2 T - 43 T^{2} - 370 T^{3} + 5041 T^{4} - 370 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 10 T + 27 T^{2} + 740 T^{3} + 11429 T^{4} + 740 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 18 T + 41 T^{2} - 1416 T^{3} - 18731 T^{4} - 1416 p T^{5} + 41 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 9 T - 43 T^{2} - 993 T^{3} - 4460 T^{4} - 993 p T^{5} - 43 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 2 T - 33 T^{2} - 820 T^{3} + 10061 T^{4} - 820 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92882001834541987631384103838, −10.35665720755647499574623760551, −10.16111542054083921994065270493, −10.05608923728147184240434983330, −9.956435109683677422448013163879, −9.311649588162327759794256093831, −8.993629421739908416521117933153, −8.747487449512283935268750637188, −8.656260700052902557464099234712, −8.105639590580036310472066482838, −7.64740172922137002343071574524, −7.23810600244196575009582988232, −6.96459900789652105320208896956, −6.93577447550883033588940439399, −6.21666430712940439343330207693, −6.15541113893471457139290388990, −5.52587764509741470347026758162, −5.47576575030228824590906326445, −4.94157221680400257439315161765, −4.70036665276223795549786942922, −4.10086683798978184237670281144, −3.08850976434554025446558559904, −2.59762376004923137830598093600, −2.14907754764985838790991192471, −1.81840793571778125165229576876,
1.81840793571778125165229576876, 2.14907754764985838790991192471, 2.59762376004923137830598093600, 3.08850976434554025446558559904, 4.10086683798978184237670281144, 4.70036665276223795549786942922, 4.94157221680400257439315161765, 5.47576575030228824590906326445, 5.52587764509741470347026758162, 6.15541113893471457139290388990, 6.21666430712940439343330207693, 6.93577447550883033588940439399, 6.96459900789652105320208896956, 7.23810600244196575009582988232, 7.64740172922137002343071574524, 8.105639590580036310472066482838, 8.656260700052902557464099234712, 8.747487449512283935268750637188, 8.993629421739908416521117933153, 9.311649588162327759794256093831, 9.956435109683677422448013163879, 10.05608923728147184240434983330, 10.16111542054083921994065270493, 10.35665720755647499574623760551, 10.92882001834541987631384103838
